Method and system for noise measurement with combinable subroutines for the mesurement, identificaiton and removal of sinusoidal interference signals in a noise signal

ABSTRACT

A method combines, within a system for noise measurement, subroutines for the measurement, identification and removal of sinusoidal interference signals (A k ·e j(ω     k     t+φ     k     ) , A k ·e j(μ·ω     k     Δt+φ     k     ) ) in a noise signal (w(t), w(ν·Δt)). In this context, the procedural stage (S 10 , S 110 , S 220 ) of splitting the frequency range (ν) to be measured into several frequency bands (ν) via an FFT filter bank ( 1 ) and the procedural stage (S 20 , S 120 , S 230 ) of determining autocorrelation matrices ({circumflex over (R)} ν ) respectively associated with the frequency bands (ν) are implemented jointly for the respectively selected subroutines of measurement, identification and removal of sinusoidal interference signals (A k ·e j(ω     k     t+φ     k     ) , A k ·e j(μ·ω     k     Δt+φ     k     ) ) in a noise signal (w(t), w(ν·Δt)). The parameters of the autocorrelation matrices ({circumflex over (R)} ν ) are adjusted in a variable manner dependent upon the respectively selected subroutine(s) and upon the required result quality.

The invention relates to a method and a system for noise measurementwith combinable subroutines for the measurement, identification andremoval of sinusoidal interference signals in a noise signal.

As illustrated in FIG. 1, measured noise signals conventionally comprisesuperimposed sinusoidal interference signals. The sources of thesesinusoidal interference signals are to be found either internally withinthe device or externally. Some of the frequencies and interference linesare known in advance (network hum up to 10 harmonics, subsidiary linesof an internal frequency synthesiser, crosstalk from frequency dividers,microphone effects, for example, from fans) and some must still bedetermined.

These spectral lines associated with sinusoidal interference signals canbe disturbing, for example, in the measurement of phase noise and musttherefore be identified and removed as well as possible from the noisemeasuring curve. However, in some applications, it is also importantjust to measure the frequencies and power levels of the sinusoidalinterference signals as accurately as possible or to compare them withknown frequencies and power levels of sinusoidal, reference interferencesignals within the framework of a reference measurement.

In the context of a high-precision spectral analysis of a measuredsignal, it is therefore desirable to identify the individual spectrallines associated with the sinusoidal interference signals from the otherspectral components of the measured signal, to measure the frequenciesand power level of the identified spectral lines and, if required, forexample, in the case of a noise measurement, to remove the identifiedspectral lines from the measured spectrum completely.

According to the prior art, graphic methods are used for theidentification of sinusoidal interference signals in a noise signal.

In one graphic method, as shown in FIG. 2, a threshold-value line isplaced over the noise curve. All of the components of the spectrumdisposed above this threshold-value line are recognised as spectrallines. The left and right intersection of the peak lines identifiedrespectively by the threshold-value line with the peak-free noise curveis determined and linked by means of an appropriate interpolation.

The identification of spectral lines associated with sinusoidalinterference signals in a noise spectrum requires a high-resolutionFourier Transform. Spectral lines disposed close together cannot beidentified separately from one another using graphic means. The use of aconstant threshold-value line is not appropriate in the case of aphase-noise curve, which provides a monotonously descending course.Consequently, a threshold-value line must be used, which provides acourse, which is constant only in very small regions, or a course whichis complementary to the phase noise curve. In such a case, measuring theposition of spectral lines is disadvantageously still only possible in aheuristic manner. A further disadvantage of the graphic method is thefact that, because of the interpolation of the noise curve, the preciseinformation about the noise curve in the region of the identifiedspectral lines is lost. Furthermore, with numerous spectral lines in thenoise curve, the graphic imprint of the spectrum curve is destroyed as aresult of the numerous interpolations. Finally, the graphic methoddisadvantageously also does not allow a separation between theindividual subroutines of measurement, identification, nor does ittherefore allow a selection of one or more of these subroutinesdependent upon the respective requirements of the measuring task.

The invention is therefore based upon the object of providing a methodand a system for the identification and/or removal of sinusoidalinterference signals in a noise signal, which is based upon an efficientnumerical method using a Fast Fourier Transform with a practicablefrequency resolution and which allows a selection of one or more of thesubroutines for noise measurement, identification and removal dependentupon the measurement task.

The object of the invention is achieved by a method with thesubroutines, which can be combined in a system for noise measurement,for the measurement, identification and removal of sinusoidalinterference signals in a noise signal according to claim 1, a systemfor noise measurement with the combinable subroutines for themeasurement, identification and removal of sinusoidal interferencesignals in a noise signal according to claim 9. Advantageous furtherembodiments of the invention are specified in the respective dependentclaims.

With the subroutine according to the invention for the identification ofsinusoidal interference signals in a noise signal, the entire frequencyrange of the noise signal to be measured is subdivided via a FastFourier Transform filter bank into several frequency bands, in which therespective noise signal is approximately white and additionally containsonly a limited, comparatively small number of spectral lines associatedwith sinusoidal interference signals.

Accordingly, the conditions are fulfilled for a determination of thefrequencies and power levels of the sinusoidal interference signals,using a method for eigenvalue analysis of autocorrelation matrices,which are obtained from the measured signal comprising the noise signaland superimposed sinusoidal interference signals. In this context, theeigenvalues for the autocorrelation matrix obtained for each frequencyband are analysed respectively into the eigenvalues associated with thenoise components and the eigenvalues associated with the signalcomponents. The eigenvalues associated with the noise components are therespectively lowest eigenvalues and provide a significantvalue-difference relative to the eigenvalues associated with the signalcomponents. Using the eigenvectors associated with the noiseeigenvalues, an estimation function is formed, of which the maximarepresent the power levels for the sought sinusoidal interferencesignals associated with the respective frequencies.

By way of example, the estimation function based upon the MUSIC(Multiple Signal Classification) method can be used as the estimationfunction. However, other frequency-estimation methods based upon theanalysis of eigenvalues of autocorrelation matrices can also be used asan alternative.

A Fast Fourier Transform filter bank is used to generate frequency bandswith a frequency bandwidth, in which the noise signal can be assumed tobe white.

On the basis of the windowing of the Fast Fourier Transform filter bank,which is realised, for example, via a Chebyshev filter, the spectrallines of sinusoidal interference signals appear not only in onefrequency band, but also in several adjacent frequency bands on the leftand the right. These additional spectral lines are undesirable; theymust be identified as such and must therefore be taken intoconsideration in the identification and removal of the spectral lines.

On the basis of the window function, the spectral lines of sinusoidalinterference signals extend over several frequency bands. With a scalingaccording to the invention of the frequency difference of each of thesespectral lines generated from the window function in the respectivefrequency band to the mid-frequency of the respective frequency bandwith the bin width of the respective FFT used, it is comparativelysimple, by deconvolution according to the invention, to re-combine thepower levels determined via the estimation function and associated withthe individual interference spectral lines, which are distributed overseveral frequency bands.

If two or more sinusoidal interference signals have a frequency, whichis disposed respectively in two or more adjacent frequency bands, and atthe same time, have the same scaled frequency on the basis of theirfrequency position relative to the mid-frequency of the respectivefrequency band, and accordingly come to be disposed at the samefrequencies over several adjacent frequency bands, the power levels ofthe spectral lines associated with the individual sinusoidalinterference signals can be separated according to the invention bymeans of a deconvolution. As an important condition for the use of theWelch method associated with the prior art, the individual FFTs of theFast Fourier Transform filter bank according to the invention areprovided with a mutual time offset in the time domain.

With the subroutine according to the invention for the removal ofsinusoidal interference signals from a noise signal, in exactly the samemanner as in the subroutine for the identification of sinusoidalinterference signals in a noise signal, the entire frequency range ofthe measured signal is divided into several frequency bands by means ofa Fast Fourier Transform filter bank. The frequencies of the sinusoidalinterference signals, which are determined either through the subroutineaccording to the invention for the identification of sinusoidalinterference signals in a noise signal described above or which arespecified by the user, are assigned to the individual frequency bands.In this context, it is also taken into consideration that, spectrallines from a sinusoidal interference signal occur not only in a singlefrequency band, but in several adjacent frequency bands as a result ofthe windowing of the FFT.

Starting from the number of spectral lines to be removed per frequencyband, the noise power of the respective frequency band is determinedaccording to the invention from a given number of lowest eigenvalues ofan autocorrelation matrix associated with the respective frequency band.The number of lowest eigenvalues of the autocorrelation matrixassociated with the respective frequency band, which are associated withthe individual noise components of the measured signal comprising thenoise signal and the sinusoidal interference signals, is determined,respectively via an analysis of all eigenvalues of the autocorrelationmatrix associated with the respective frequency band, in that all of theeigenvalues associated with the respective noise components are minimaland provide a significant value difference relative to the signalcomponents—the sinusoidal interference signals. With a specified numberof interference-signal spectral lines to be removed, the number ofeigenvalues associated with the noise signal can, according to theinvention, also be determined from the matrix dimension minus thespecified number of interference-signal spectral lines to be removed.The spectrum of the noise signal after the removal of the sinusoidalinterference signals is derived according to the invention from thecombination of all of the noise spectra associated with the individualfrequency bands.

Since the two subroutines for the identification and removal ofsinusoidal interference signals in a noise signal respectively containthe procedural stages of splitting the frequency range via an FFT filterbank into several frequency bands and determining the autocorrelationmatrices associated with the respective frequency bands, and, as will beshown in greater detail below, the subroutine for noise measurement canalso be realised via these two procedural stages, these two proceduralstages according to the invention are implemented jointly for all threesubroutines.

In this context, it must be taken into consideration that the matrixdimension of the autocorrelation matrices is of a different size foreach of the three subroutines. Accordingly, in a first embodiment of themethod and the system according to the invention for noise measurementwith the combinable subroutines for the measurement, identification andremoval of sinusoidal interference signals in a noise signal, therespective maximum matrix dimension is used for each of theautocorrelation matrices. In this embodiment, the computational costdoes, in fact, represent a maximum, however, evaluation data areavailable for processing at all times for all three subroutines withoutinterruption. In a second embodiment of the method and the systemaccording to the invention, in the normal operating mode, theautocorrelation matrix is calculated with the minimum matrixdimension—the matrix dimension required for noise measurement at thelevel of the value 1—and, in the case of an implementation of thesubroutine for identification or removal, the respective autocorrelationmatrix is re-initialised with the matrix dimension required for therespective subroutine for identification or removal. Accordingly, in thecase of the second embodiment, each subroutine is implemented at itsrespectively maximum rate; however, in the subroutine for identificationor removal, it is not possible to access the evaluation data from beforethe time of initialisation, because of the re-initialisation of theautocorrelation matrix.

Alongside the parameter of the matrix dimension for the respectiveautocorrelation matrices, the parameter of the number of averagings canalso be varied according to the invention within the framework of are-initialisation of the autocorrelation matrices. In the case of alarge noise variance of the measured signal, sinusoidal interferencesignals, which are difficult to identify, can advantageously beidentified in the noise spectrum with less ambiguity and greatersecurity through an additional averaging up to the present time.

The two embodiments of the method and system according to the inventionfor noise measurement with the combinable subroutines for themeasurement, identification and removal of sinusoidal interferencesignals in a noise signal are explained in greater detail below withreference to the drawings. The drawings are as follows:

FIG. 1 shows a spectral display of a measured spectrum of a noise signalwith superimposed interference-signal spectral lines;

FIG. 2 shows a spectral display of a noise spectrum with theinterference-signal spectral lines removed on the basis of graphicmethods;

FIG. 3 shows a block diagram of a Fast Fourier Transform filter bank;

FIGS. 4A, 4B show a frequency spectrum of a Chebyshev window and arectangular window;

FIG. 5 shows a frequency-time diagram of an application according to theinvention of a Fast Fourier Transform filter bank;

FIG. 6 shows a schematic display of a FFT filter bank consisting of FFTswith a time-overlap;

FIG. 7 shows a spectral display of power-level curves;

FIG. 8 shows a spectral display of the analysis range forinterference-signal spectral lines at half-decade thresholds;

FIG. 9 shows a flow chart for a subroutine according to the prior artfor the measurement of a noise-power spectrum;

FIG. 10 shows a flow chart for a subroutine according to the inventionfor the detection of sinusoidal interference signals in a noise signal;

FIG. 11 shows a flow chart for a subroutine according to the inventionfor the removal of sinusoidal interference signals from a noise signal;

FIG. 12 shows a flow chart for a first embodiment of the methodaccording to the invention for noise measurement with the combinablesubroutines according to the invention for the measurement,identification and removal of sinusoidal interference signals from anoise signal;

FIG. 13 shows a flow chart for a second embodiment of the methodaccording to the invention for noise measurement with the combinablesubroutines according to the invention for the measurement,identification and removal of sinusoidal interference signals from anoise signal;

FIG. 14 shows a block diagram of a system according to the invention fornoise measurement with the combinable functions according to theinvention for the measurement, identification and removal of sinusoidalinterference signals in a noise signal;

FIG. 15 shows a spectral display of a noise spectrum contaminated withspectral lines of sinusoidal interference signals; and

FIG. 16 shows a spectral display of a noise spectrum with spectral linesof sinusoidal interference signals removed.

Before describing the subroutine according to the invention for thedetection of sinusoidal interference signals in a noise signal, thesubroutine according to the invention for the removal of sinusoidalinterference signals from a noise signal and the method according to theinvention for noise measurement with the combinable subroutinesaccording to the invention for the measurement, identification andremoval of sinusoidal interference signals in a noise signal in greaterdetail with reference to FIGS. 9, 10, 11, 12, 13, and 14, the followingsection presents the theoretical background, on which the invention isbased.

According to the invention, the entire frequency range of the measuredsignal comprising the noise signal and sinusoidal interference signalsis split into several frequency bands, in which the respective noisesignal is approximately white and which each contain only a limited,comparatively small number of sinusoidal interference signals. Thesubdivision into several frequency bands takes place via a filter bank,which is realised in the form of a Fast Fourier Transform (FFT)according to FIG. 3.

An FFT filter bank 1, which, according to the invention, generates anumber of frequency bands corresponding to its FFT length N_(FFT),accordingly comprises a total of N_(FFT) signal paths, which, from ameasured signal x(t), leads to a total of N_(FFT) time-discrete outputsignals x₁(μ·Δt), x₂(μ·Δt), . . . , x_(NFFT)(μ·Δt). With a real measuredsignal x(t) and therefore with a symmetrical spectrum X(f), only the FFTin one sideband needs to be evaluated. In this case, the number offrequency bands to be observed is reduced to N_(FFT)/2. The input signalx(t) is subjected in each of the N_(FFT) signal paths first to awindowing 2 ₁, 2 ₂, . . . , 2 _(NFFT) with the respectively associatedimpulse response h_(Window1)(t), h_(Window2)(t), . . . ,h_(WindowNFFT)(t). The mid-frequencies of the respective windowtransmission function H_(Window1)(f), H_(Window2)(f), . . . ,H_(WindowNFFT)(f) agree with the frequencies ν·f₀ of the individual FFTbins and also form the main frequencies of the individual frequencybands ν. The bandwidth of each of the window transmission functionsH_(Window1)(f), H_(Window2)(f), . . . , H_(WindowNFFT)(f) is thereforederived as follows—under the ideal condition of rectangular frequencyspectra of the window transmission functions H_(Window1)(f),H_(Window2)(f), . . . , H_(WindowNFFT)(f)—from the bin width f₀ of theFFT and also corresponds to the bandwidth Δf of each of the frequencybands.

After the windowing 2 ₁, 2 ₂, . . . , 2 _(NFFT), each windowed signal ismixed in subsequent down mixings 3 ₁,3 ₂, . . . , 3 _(NFFT), with thefrequency f₀,2·f₀, . . . ,N_(FFT)·f₀ into the respective frequency bandin the baseband.

Finally, with the signals windowed and mixed down into the baseband, anundersampling 4 ₁,4 ₂, . . . ,4 _(NFFT) takes place in the individualsignal paths with the respective sampling rate$\frac{1}{{N_{FFT} \cdot \Delta}\quad t},$, which leads to the individual time-discrete output signals x₁(μ·Δt),x₂(μ·Δt), . . . , x_(NFFT)(μ·Δt) at the respective outputs of theindividual signal paths. Each of the time-discrete output signalsx₁(μ·Δt), x₂(μ·Δt), . . . , x_(NFFT)(μ·Δt) is assigned to one of thetotal of N_(FFT) frequency bands and each provides an approximatelywhite noise signal and has a limited, comparatively small number ofsinusoidal interference signals.

In reality, the window transmission functions H_(Window1)(f),H_(Window2)(f), . . . , H_(WindowNFFT)(f) do not provide rectangularfrequency spectra, but, with a Chebyshev filter as shown in FIG. 4A orwith a rectangular filter as shown in FIG. 4B, provide several“subsidiary lobes” disposed to the left and right of a “main lobe”.While a rectangular filter provides a narrow “main lobe” and thereforemost nearly fulfils the requirement for the realisation of ahigh-frequency resolution corresponding to the bin width f₀ of the FFT,one spectral line disadvantageously continues periodically in theadjacent “subsidiary lobes” corresponding to the attenuation of therespective “subsidiary lobe” (so-called “leakage effect”). In the caseof the Chebyshev filter, by contrast, the periodic continuation in theindividual “subsidiary lobes” of spectral lines occurring in the “mainlobe” can be reduced in a targeted manner to a negligible value byattenuating the “subsidiary lobes”, while the bandwidth of the “mainlobe” disadvantageously extends over several FFT bins and therefore overseveral frequency bands. The removal according to the invention of thedisadvantageous influence of the window function over several frequencybands is explained in greater detail below.

The output signal x(μ·Δt·N_(FFT)) in the ν-^(th) frequency bandgenerated by the FFT filter bank is derived, in the case of atime-discrete input signal x(μ·Δt), according to equation (1):$\begin{matrix}{{x\quad\left( {{\mu \cdot \Delta}\quad{t \cdot N_{FFT}}} \right)} = {\Delta\quad{t \cdot {\sum\limits_{\mu}{x\quad{\left( {{\mu\Delta}\quad t} \right) \cdot h}\quad{\left( {{\mu\Delta}\quad t} \right) \cdot {\mathbb{e}}^{{- j}\frac{2\pi}{N_{FFT}}v\quad\mu}}}}}}} & (1)\end{matrix}$

If the time-discrete input signal x(μ·Δt) contains a sinusoidalinterference signal with a frequency f_(k), which is disposed exactly onthe mid-frequency of the FFT bin (f_(k)=ν·f₀), then, withx(μΔt)=e^(j2πνf) ⁰ ^(μΔt) and the relationship${f_{0} = \frac{1}{N_{FFT}\Delta\quad t}},$an output signal x(μ·Δt·N_(FFT)) of the FFT filter bank is derivedaccording to equation (2): $\begin{matrix}\begin{matrix}{{x\quad\left( {{\mu \cdot \Delta}\quad{t \cdot N_{FFT}}} \right)} = {\Delta\quad{t \cdot {\sum\limits_{\mu}{{{\mathbb{e}}^{{j2\pi}\quad{vf}_{0}{\mu\Delta}\quad t} \cdot h}\quad{\left( {{\mu\Delta}\quad t} \right) \cdot {\mathbb{e}}^{{- j}\frac{2\pi}{N_{FFT}}v\quad\mu}}}}}}} \\{= {\Delta\quad{t \cdot {\sum\limits_{\mu}{h\quad\left( {{\mu\Delta}\quad t} \right)}}}}} \\{= {{const}.}}\end{matrix} & (2)\end{matrix}$

If the frequency f_(k) of the sinusoidal interference signal is disposedcentrally between two FFT bins$\left( {f_{k} = {{vf}_{0} \pm \frac{f_{0}}{2}}} \right),$then the output signal x(μ·Δt·N_(FFT)) of the FFT filter bank in theν-^(th) frequency band provides a rotating phasor with maximumrotational velocity $\frac{f_{0}}{2}$according to equation (3) $\begin{matrix}\begin{matrix}{{x\quad\left( {{\mu \cdot \Delta}\quad{t \cdot N_{FFT}}} \right)} = {\Delta\quad{t \cdot {\sum\limits_{\mu}{{{\mathbb{e}}^{{{j2\pi}{({{vf}_{0} \pm \frac{f_{0}}{2}})}}\quad{\mu\Delta}\quad t} \cdot h}\quad{\left( {{\mu\Delta}\quad t} \right) \cdot {\mathbb{e}}^{{- j}\frac{2\pi}{N_{FFT}}v\quad\mu}}}}}}} \\{= {\Delta\quad{t \cdot {\sum\limits_{\mu}{h\quad{\left( {{\mu\Delta}\quad t} \right) \cdot {\mathbb{e}}^{{\pm {j2\pi}}\frac{f_{0}}{2}\Delta\quad t}}}}}}}\end{matrix} & (3)\end{matrix}$

As a result of the windowing, spectral lines of sinusoidal interferencesignals, of which the frequency f_(k) comes to be disposed in a givenfrequency band, are also detected in adjacent frequency bands. Theamplitude of these subsidiary spectral lines is derived from theattenuation of the window transmission function.

According to equation (4), the phases of the subsidiary spectral linesare constant, subject to the condition of a real and symmetrical (even)window function h_(Window)(t) and therefore also of a real and evenwindow transmission function H_(Window)(f) relative to the phase of themain spectral line generated by the “main lobe” of the windowtransmission function:

(A _(k)·δ(f−f _(k))*H _(Window)(f))=

(A _(k) ·H _(Window)(f−f _(k)))=

A _(k) =const.  (4)

By way of summary, the frequency-time diagram of the FFT filter bank inFIG. 5 once again shows the connections between the time domain and thefrequency domain with an FFT filter bank according to the invention,which, via several Fast Fourier transformers FFT₁, FFT₂, FFT₃, FFT₄ andFFT₅, generates a total of N_(FFT)=8 frequency bands, which correspondrespectively to the individual signal paths of the FFT filter bank inFIG. 3, with a bandwidth at the level of the bin width f₀ of the FFTs.

The individual Fast Fourier transformers FFT₁, FFT₂, FFT₃, FFT₄ and FFT₅of the FFT filter bank need not be connected cyclically one after theother, but can also provide a time overlap as shown in FIG. 6. Thisarrangement corresponds to the Welch method known from the prior art.The respective N_(FFT) sampling values at the outputs of the respectiveN_(FFT) signal paths of the FFT filter bank 1 are present according toFIG. 3 at the positions x₁(μ·Δt·N_(FFT)·(1−overlap)),x₂(μ·Δt·N_(FFT)·(1−overlap)), . . . ,x_(NFFT)(μ·Δt·N_(FFT)·(1−overlap)).By comparison with the non-overlapping FFTs, there is therefore anoversampling by the factor $\frac{1}{1 - {overlap}}.$

Additionally, the overlap in the case of a sinusoidal interferencesignal with a frequency f_(k), which provides a frequency difference Δf₀from the mid-frequency of the FFT bin (f_(k)=ν·f₀+Δf₀), brings about arotating phasor, which provides a faster rotational velocity$\frac{\Delta\quad f_{0}}{1 - {overlap}}.$

The overlapping of the individual FFT window of the FFT filter bank 1leads to a correlation between the values of the measured signal. As aresult of the overlap, oversampled FFT results are obtained. Theoverlapping factor ov is obtained with an overlap according to equation(5): $\begin{matrix}{{ov} = \frac{1}{1 - {overlap}}} & (5)\end{matrix}$

Only FFT results with a difference of ov or more are formed fromnon-overlapping FFT windows of the FFT filter bank 1. The oversamplingis compensated by an undersampling by a factor $\frac{1}{1 - {overlap}}$in the individual FFTs. In this manner, un-correlated noise samplingvalues, and therefore also a white noise signal necessary for theeigenvalue analysis of autocorrelation matrices, is provided.

If only an undersampling by the factor N_(FFT) is implemented in theindividual undersamplings of the FFTs of the FFT filter bank, as shownin FIG. 3, then a data sequence x_(ν)(μ)=x_(ν)(μ·Δt) is obtained at theoutput of the FFTs for the frequency band ν as shown in equation (6),wherein w(μ) models the noise, and the summated term models the total ofp sinusoidal interference signals: $\begin{matrix}{{x_{v}(\mu)} = {{w\quad(\mu)} + {\sum\limits_{k = 1}^{p}{A_{k} \cdot {\mathbb{e}}^{j\quad{({{\mu\omega}_{k} + \varphi_{k}})}}}}}} & (6)\end{matrix}$

An undersampling by the factor N_(FFT)(1−overlap) in the individual FFTsof the FFT filter bank 1 leads to a non-oversampled data sequencex₁(μ·ov) according to equation (7) only for every ov-^(th) value:$\begin{matrix}{{x_{v}\left( {\mu \cdot {ov}} \right)} = {{w\left( {\mu \cdot {ov}} \right)} + {\sum\limits_{k = 1}^{P}{A_{k} \cdot {\mathbb{e}}^{j{({{\mu \cdot {ov} \cdot \omega_{k}} + \varphi_{k}})}}}}}} & (7)\end{matrix}$

According to equation (8), the scaled angular frequency ω_(norm,k) isthe non-oversampled angular frequency ω_(k) scaled to the bin width f₀of the FFT filter bank of the spectral line associated with a sinusoidalinterference signal and represents the frequency difference Δf₀ of thespectral line at the frequency ν·f₀ of the nearest FFT bin, whichcorresponds to the mid-frequency ν·f₀ of the respective frequency band.$\begin{matrix}{\omega_{{norm},k} = {{ov} \cdot \frac{2\pi\quad f_{k}}{f_{0}}}} & (8)\end{matrix}$

The scaled angular frequency ω_(norm,k) therefore provides a value range[−π,+π]. At ω_(norm,k)=±π, the associated spectral line is disposedexactly at the right-hand or left-hand edge of the respective FFT binand can also be found with the same strength in the adjacent frequencyband.

According to equation (9) the ideal autocorrelation matrix R, whichprovides, for example, the dimension M×M, is obtained from anon-oversampled output signal x_(ν)(μ·ov) at the output of the FFTassociated with the frequency band ν. Sampled values disposed at a pasttime of the non-oversampled output signal x_(ν)(μ·ov) are used for thispurpose:R=E{x·x ⁺ }mit x=[x(μ),x(μ−ov), . . . ,x(μ−M·ov)]^(T)  (9)[mit=with]

The autocorrelation matrix R is a square, positive and definite matrix,that is to say, its eigenvalues are real and positive. The eigenvectors,associated with the non-equal eigenvalues, are also orthogonal.

As a result of the stochastic—noise-laden—character of thenon-oversampled output signal x_(i)(μ·ov) at the output of the FFTfilter bank 1, a reliable estimate for the autocorrelation matrix{circumflex over (R)}_(ν) according to equation (10) is determinedthrough multiple averaging—a total of N_(avg) times: $\begin{matrix}{{\hat{R}}_{v} = {\frac{1}{N_{avg}}{\sum\limits_{\mu = {{({M - 1})} \cdot {ov}}}^{{{({M - 1})} \cdot {ov}} + N_{avg} - 1}{\begin{bmatrix}{x(\mu)} \\{x\left( {\mu - {ov}} \right)} \\\vdots \\{x\left( {\mu - {\left( {M - 1} \right) \cdot {ov}}} \right)}\end{bmatrix} \cdot {\quad\left\lbrack {{x^{*}(\mu)},{x^{*}\left( {\mu - {ov}} \right)},\ldots\quad,{x^{*}\left( {\mu - {\left( {M - 1} \right) \cdot {ov}}} \right)}} \right\rbrack}}}}} & (10)\end{matrix}$

According to equation (11), the dimension M of the estimate ({circumflexover (R)}_(ν)) of the autocorrelation matrix associated with thefrequency band (ν) for the identification according to the invention ofsinusoidal interference signals in a noise signal must correspond atleast to the maximum expected number p_(max) of spectral lines perfrequency band ν with the addition of the value 2. The value 2 resultsfrom the fact that the MUSIC method requires at least two noiseeigenvalues.M≧p _(max)+2  (11)

In the special case, in which the matrix dimension M similarly adoptsthe value 1, starting from equation (10), the Welch method, known fromthe prior art for the calculation of the noise spectrum in the case oftime-overlapping FFTs, is derived according to equation (12) for theautocorrelation matrix ({circumflex over (R)}_(ν)). $\begin{matrix}{{\hat{R}}_{v} = {{\frac{1}{N_{avg}} \cdot {\sum\limits_{k = 0}^{N_{avg} - 1}{{x_{v}(k)} \cdot {x_{v}^{*}(k)}}}} = {\hat{S}\left( {v \cdot f_{0}} \right)}}} & (12)\end{matrix}$

According to the MUSIC (Multiple Signal Classification) method, the Meigenvalues λ₁, . . . ,λ_(M) associated with the estimated value{circumflex over (R)}_(ν) of the autocorrelation matrix and theassociated eigenvectors v ₁, . . . ,v _(M) of the respective frequencyband ν are calculated after a total of N_(avg) averaging stagesaccording to equation (10).

The eigenvalues of the autocorrelation matrix can be subdivided into twogroups. The first group of the lowest eigenvalues, of which the numberM−p is equal to the dimension M of autocorrelation matrix {circumflexover (R)}_(ν) reduced by the number p of sinusoidal interference signalspresent in the frequency band ν, are associated with the noisecomponents of the measured signal x(n·ov). The second group of theremaining eigenvalues, which are associated with the signalcomponents—in the present case with the sinusoidal interferencesignals—and of which the number therefore corresponds to the number psinusoidal interference signals, each provide a significant valuedifference relative to each of the respective lowest eigenvaluesassociated with the noise components.

No deterministic method exists for separating all eigenvalues λ₁, . . .,λ_(M) of the estimate {circumflex over (R)}_(ν) of the autocorrelationmatrix associated with the frequency band ν into the first and secondgroup of eigenvalues and therefore for the determination of the number pof sinusoidal interference signals in the measured signal x(n·ov).According to the prior art, only a statistical analysis of theeigenvalues by means of histogram can be used in order to separate theeigenvalues λ₁, . . . ,λ_(M).

In the case of an ideal autocorrelation matrix R, the lowest eigenvaluesλ₁, . . . ,λ_(M−p) associated with the noise components are identicaland, according to equation (13), equal to the noise power σ_(w) ².σ_(w) ²=λ_(i) for iε[1, . . . ,M−p]  (13)

In the case of a numerically-estimated autocorrelation matrix{circumflex over (R)}_(i), the noise eigenvalues λ₁, . . . ,λ_(M−p) aredistributed about the actual noise power σ_(w) ² as a mean value. Inthis case, the noise power σ_(w) ² is determined according to equation(14): $\begin{matrix}{\sigma_{w}^{2} = {\frac{1}{M - p}{\sum\limits_{k = 1}^{M - p}\lambda_{k}}}} & (14)\end{matrix}$

The variance σ of the noise eigenvalues according to equation (15)declines with an increasing averaging duration N_(avg). $\begin{matrix}{\sigma = \sqrt{\frac{1}{k}{\sum\limits_{i = 1}^{k}{{\lambda_{i} - \mu}}^{2}}}} & (15)\end{matrix}$

The scaled angular frequencies ω_(norm,k) of the spectral linesassociated with the individual sinusoidal interference signals of afrequency band ν are determined with the assistance of an estimationfunction, which is based upon the eigenvalue analysis of autocorrelationmatrices. These methods, which form the prior art, are described indetail in Manon H. Hayes: “Statistical digital signal processing andmodelling”, John Wiley & sons. Inc., 1996, page 451 to 469. The MUSIC(Multiple Signal Classification) method and respectively the Root MUSICmethod are presented here briefly, without going into the details, onlyby way of example.

With the pure MUSIC method, on the basis of the eigenvectors v _(i)associated with the noise components and an arbitrary column vectore(ω_(normk)) of the signal correlation matrix R_(s) according toequation (16), an estimation function P_(MU)(e^(jω) ^(normk) ) accordingto equation (17) is formed:e (ω_(normk))=[1e ^(jω) ^(normk) ,e ^(j2ω) ^(normk) , . . . ,e^(j(M−1)ω) ^(normk) ]  (16) $\begin{matrix}{{P_{MU}\left( {\mathbb{e}}^{{j\omega}\quad{normk}} \right)} = \frac{1}{\sum\limits_{i = 1}^{M - p}{{{{\underset{\_}{e}}^{+}\left( \omega_{normk} \right)} \cdot {\underset{\_}{v}}_{i}}}^{2}}} & (17)\end{matrix}$

As shown in detail in Manon H. Hayes: “Statistical digital signalprocessing and modelling”, John Wiley & sons. Inc., 1996, page 451 to469, each of the eigenvalues v _(i) associated with the individual noisecomponents is orthogonal to an arbitrary column vector e(ω_(normk)) ofthe signal autocorrelation matrix R_(s) with a scaled angular frequencyω_(norm,k) of a sinusoidal interference signal in the non-oversampledoutput signal x_(i)(n·ov) at the output of the FFT filter bank 1. Inthis case, the scalar product e(ω_(normk))·v _(i) is zero, andaccordingly, the estimation function P_(MU)(e^(jω) ^(normk) ) is at amaximum. The scaled angular frequencies ω_(norm,k) at the respective plargest maxima of the estimation function P_(MU)(e^(jω) ^(normk) )therefore represent the scaled angular frequencies ω_(norm,k) of thesinusoidal interference signals in the non-oversampled output signalx_(i)(n·ov) at the output of the FFT filter bank 1. In this manner, withthe estimation function P_(MU)(e^(jω) ^(normk) ), an estimation functionfor the identification of the scaled angular frequencies ω_(norm,k) ofthe sinusoidal interference signals in the non-oversampled output signalx_(i)(n·ov) is obtained at the output of the FFT filter bank 1.

The calculation of the estimation function P_(MU)(e^(jω) ^(normk) ) viathe angular frequency ω_(norm,k), which represents a pseudo spectrum,can be advantageously implemented via an FFT. In this context, the FFTlength determines the frequency resolution of the calculated estimationfunction P_(MU)(e^(jω) ^(normk) ).

In the case of the Root MUSIC Method, the Z-transform V_(i)(z) of theindividual eigenvectors v_(i) respectively associated with the noisecomponents is determined. For this purpose, the individual componentsv_(i)(l) of the respective noise eigenvectors v_(i) are subjected to aZ-transformation according to equation (18): $\begin{matrix}{{{V_{i}(z)} = {{\sum\limits_{l = 0}^{M - 1}{{{v_{i}(k)} \cdot^{- l}\quad{for}}\quad{all}\quad i}} = 1}},\ldots\quad,{M - p}} & (18)\end{matrix}$

The estimates for the total of p scaled angular frequencies ω_(norm,k)of the sinusoidal interference signals result from the angles of the pzero points of the polynomial D(z) calculated from the Z-transformV_(i)(z) according to equation (19), which are disposed nearest on theunit circle of the complex z-plane: $\begin{matrix}{{D(z)} = {\sum\limits_{i = 1}^{M - p}{{V_{i}(z)}{V_{i}^{*}\left( {1/z^{*}} \right)}}}} & (19)\end{matrix}$

The scaled angular frequencies ω_(normk) can only be determined with anaccuracy of modulus 2π, both with the pure MUSIC method via the FFTcalculation of the pseudo spectrum and also with the Root-MUSIC methodusing the zero-point search on the unit circle of the complex z-plane,so that, in particular, the scaled angular frequencies ω_(normk) at theedges of the respective frequency band cannot be identifiedunambiguously. An unambiguous identification can only be achieved bycombining the results determined in the individual frequency bands.

The power levels P_(MUi,k) of the respective p sinusoidal interferencesignals at the scaled angular frequencies ω_(normk) of a frequency bandν are obtained from the solution of the equation system (20), as is alsoshown in detail in Manon H. Hayes: “Statistical digital signalprocessing and modelling”, John Wiley & sons. Inc., 1996, page 459 to463, $\begin{matrix}{{{\sum\limits_{k = 1}^{p}{P_{{MUi},k} \cdot {{V_{i}\left( {\mathbb{e}}^{{j\omega}_{normk}} \right)}}}} = {{\lambda_{i} - {\sigma_{w}\quad{for}\quad{all}\quad i}} = 1}},\ldots\quad,p} & (20)\end{matrix}$

The Z-transform V_(i)(e^(jω) ^(normk) ) of the eigenvectors v _(i)associated with the respective p interference signal components areobtained from equation (21) with reference to equation (18) for the M−pnoise-component eigenvectors: $\begin{matrix}{{{V_{i}\left( {\mathbb{e}}^{{j\omega}_{k}} \right)} = {{\sum\limits_{l = 0}^{M - 1}{{v_{i}(l)}\quad{\mathbb{e}}^{{- {j\omega}_{normk}}l}\quad{for}\quad{all}\quad i}} = 1}},\ldots\quad,p} & (22)\end{matrix}$

The eigenvectors λ_(M−p+1), λ_(M−p+2), . . . ,λ_(M) associated with thep sinusoidal interference signals are the largest eigenvectors, arrangedrespectively in ascending order, of the estimated value {circumflex over(R)}_(ν) of the autocorrelation matrix for the frequency band ν. σ_(w)is the noise power predominating in the respective frequency band ν. Thevectorial presentation of equation system (20) is derived from equationsystem (22): $\begin{matrix}{\quad{\begin{bmatrix}{{V_{M - p + 1}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 1}} \right)}}^{2} & {{V_{M - p + 1}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 2}} \right)}}^{2} & \cdots & {{V_{M - p + 1}\left( {\mathbb{e}}^{{j\omega}_{normp}} \right)}}^{2} \\{{V_{M - p + 2}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 1}} \right)}}^{2} & {{V_{M - p + 2}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 2}} \right)}}^{2} & \cdots & {{V_{M - p + 2}\left( {\mathbb{e}}^{{j\omega}_{normp}} \right)}}^{2} \\\vdots & \vdots & ⋰ & \vdots \\{{V_{M}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 1}} \right)}}^{2} & {{V_{M}\left( {\mathbb{e}}^{{j\omega}_{{norm}\quad 2}} \right)}}^{2} & \cdots & {{V_{M}\left( {\mathbb{e}}^{{j\omega}_{normp}} \right)}}^{2}\end{bmatrix}{\quad\quad}{\quad{\quad{\begin{bmatrix}P_{{MUv},1} \\P_{{MUv},2} \\\vdots \\P_{{MUv},p}\end{bmatrix} = \begin{bmatrix}{\lambda_{M - p + 1} - \sigma_{w}^{2}} \\{\lambda_{M - p + 2} - \sigma_{w}^{2}} \\\vdots \\{\lambda_{M} - \sigma_{w}^{2}}\end{bmatrix}}}}}} & (22)\end{matrix}$

After the power level P_(MUν,k) and the scaled angular frequenciesω_(normk) of the respective p(ν) sinusoidal interference signals havebeen determined for every frequency band ν, for example, using the MUSICmethod, as presented above, the results of the individual frequencybands ν must be combined to form a combined result. In this context, itmust be taken into consideration that spectral lines of sinusoidalinterference signals are obtained in adjacent frequency bands because ofthe windowing of the FFT filter bank 1.

The problem of frequency-band-wide superimposition is already known fromthe prior art in the context of the time overlap of several FFTs(overlapped FFT). The time overlap in this context serves to compensatethe non-constant window-transmission function with regard to anapproximately constant evaluation of all spectral values in the entirefrequency range. According to the prior art, an estimation of thefrequency spectrum generated through an FFT filter bank 1 withoverlapping in the time domain is realised by means of the Welch method.

As shown in detail in Manon H. Hayes: “Statistical digital signalprocessing and modelling”, John Wiley & sons. Inc., 1996, page 415 to420, the expected value E{Ŝ_(Welch)(e^(jω))} of a frequency spectrumgenerated from several time-overlapped FFTs according to equation (23)is derived from the convolution of the frequency spectrum S(e^(jω))generated through an FFT filter bank 1 with the modulus-squared windowtransmission function H(e^(jω)):E{Ŝ _(Welch)(e ^(jω))}=S(e ^(jω))*|H(e ^(jω))|²  (23)

The variance of a frequency spectrum of this kind generated via theWelch method tends toward zero with an increasing averaging length.

The above Welch method is used according to the invention, to achieve anunambiguous identification of main and subsidiary lines within one ofthe frequency bands ν, which can be assigned to several sinusoidalinterference signals with frequencies in different frequency bands. Theindividual frequency bands are overlapped in the time domain for thispurpose as shown in FIG. 6.

The expected value for a frequency spectrum of an FFT filter bank 1consisting of several spectral lines is derived starting from equation(23) using the Welch method according to equation (24): $\begin{matrix}{{E\quad\left\{ {{\hat{S}}_{Welch}\left( {\mathbb{e}}^{j\omega} \right)} \right\}} = {{\sum\limits_{k}{\left( {{P_{k} \cdot \delta}\quad\left( {\omega - \omega_{k}} \right)} \right)*{{H\left( {\mathbb{e}}^{j{(\omega)}} \right)}}^{2}}} = {\sum\limits_{k}{P_{k} \cdot {{H\left( {\mathbb{e}}^{j{({\omega - \omega_{k}})}} \right)}}^{2}}}}} & (24)\end{matrix}$

Since the frequency spectrum generated by the FFT filter bank 1 and thenmodified using the Welch method is only calculated at the discretefrequencies f₀ of the FFT bins, the expected value of a frequencyspectrum of an FFT filter bank 1 consisting of several spectral lines isobtained according to equation (25): $\begin{matrix}{{E\quad\left\{ {{\hat{S}}_{Welch}\left( {\mathbb{e}}^{j\quad v\quad 2\quad\pi\quad f_{0}} \right)} \right\}} = {\sum\limits_{k}{P_{k} \cdot {{H\left( {\mathbb{e}}^{j{({{v\quad 2\pi\quad f_{0}} - {2\pi\quad f_{k}}})}} \right)}}^{2}}}} & (25)\end{matrix}$

For the frequencies f_(k) of the individual spectral lines, therelationship in equation (26) applies, in which the frequency f_(k) ofthe spectral line is classified in the FFT frequency grid by specifyingthe frequency offset Δf₀ relative to the nearest FFT bin.$\begin{matrix}{f_{k} = {{{v \cdot f_{0}} + {\Delta\quad f_{ok}}} = {{v \cdot f_{0}} + {\frac{\omega_{norm}}{2\pi}f_{0}}}}} & (26)\end{matrix}$

Accordingly, for every scaled angular frequency ω_(normk), there is anindividual expected value Ŝ_(Welch)(e^(jν2πf) ⁰ ) for the frequencyspectrum at the discrete frequency f₀ of the FFT bin in the frequencyband ν.

Accordingly, the power level P_(MUν), determined, for example, by meansof the MUSIC method, of a spectral line at the scaled angular frequencyω_(normk) in the frequency band ν is derived according to equation (27)as the expected value Ŝ_(Welch)(e^(jν2πf) ⁰ ) for the frequencyspectrum, which is calculated from the sum of all respective powerlevels P_(k) multiplicatively linked with the modulus-squared, windowtransmission function frequency-displaced by the frequency f_(k) withapproximately identical scaled angular frequency ω_(normk) andaccordingly identical frequency offset Δf₀ relative to the respectiveFFT bin frequency: $\begin{matrix}{P_{MUv} = {{E\quad\left\{ {{\hat{S}}_{Welch}\left( {\mathbb{e}}^{j\quad{v2}\quad\pi\quad f_{0}} \right)} \right\}} = {\sum\limits_{k}{P_{k} \cdot {{H\left( {\mathbb{e}}^{j{({{v\quad 2\pi\quad f_{0}} - {2\pi\quad f_{k}}})}} \right)}}^{2}}}}} & (27)\end{matrix}$

All spectral lines, of which the scaled angular frequencies ω_(normk)provide an angular-frequency difference Δω_(normk) according to equation(28), which is smaller than a maximum angular-frequency differenceΔω_(normMax), are counted as spectral lines in different frequency bandsν with approximately identical scaled angular frequency ω_(normk).Δω_(normk)<Δω_(normMax)

  (28)

The power levels P_(MUν) at the individual FFT bin frequencies ν·f₀,which can be calculated for each given scaled angular frequencyω_(normk) according to equation (28), are obtained in sum for each givenscaled angular frequency ω_(normk) in a power-level curve according toFIG. 7.

The number p(ν) of sinusoidal interference signals, the scaledfrequencies ω_(normk) and the power levels P_(MUν) of the respectivespectral lines associated with the p(ν) sinusoidal interference signalsare derived using the MUSIC method and the subsequent Welch method foreach frequency band ν. Ambiguity continues to exist with regard to thefrequencies f_(k) and the power level P_(k) of those spectral lines,which contribute to the power-level curve with approximately identicalscaled angular frequency ω_(normk) in adjacent frequency bands ν at therespective FFT bin frequency ν·f₀.

For the determination presented below of the individual power levelsP_(k) of those spectral lines, which contribute to the power-level curvewith approximately identical scaled angular frequency ω_(normk) inadjacent frequency bands ν at the respective FFT bin frequency ν·f₀, aweighted, scaled angular frequency {overscore (ω)}_(normk) according toequation (29) is introduced in the individual frequency bands ν insteadof the scaled angular frequency ω_(normk): $\begin{matrix}{{\overset{\_}{\omega}}_{normk} = {\sphericalangle{\sum\limits_{v}{P_{MUv} \cdot e^{{j\omega}_{normkv}}}}}} & (29)\end{matrix}$

By taking into consideration the power levels P_(MUν) in the weighted,scaled angular frequency {overscore (w)}_(normk), the scaled angularfrequencies ω_(normkν) of spectral lines, which provide a higher powerlevel P_(MUν), are more heavily weighted. Through the use of the complexexponential function e^(jω) ^(normkν) for the scaled angular frequencyω_(normν), in particular, at the frequency band edges (ω_(normkν)=±π),the value of the scaled angular frequency ω_(normkν) is preserved in theaveraging.

Starting from equation (27), the relationship between the power levelsP_(MUν) determined, for example, using the MUSIC method in the frequencyband ν at a given weighted, scaled angular frequency {overscore(w)}_(normk) and the linear combination of the sought power levels P_(k)of spectral lines, which result from sinusoidal interference signalswith angular frequencies ω_(k) in adjacent frequency bands ν±i and whichare superimposed at the power level P_(MUν) of the power-level curve inthe frequency band ν, is described by the equation system (30). Theangular frequencies ω_(k) of the main lines and subsidiary linesassociated respectively with a sinusoidal interference signal, which areassociated in sum with a common power-level curve, are disposedrespectively in adjacent frequency bands ν±i and all provide theidentical frequency difference Δf₀ at the respective frequency bandmid-frequency or respective FFT bin frequency (ν±i)·f₀. The power-levelcurves begin respectively in the frequency band n_(Start) and extendover a total of N_(LP) frequency bands. In the equation system (30), itis assumed that a main line of a sinusoidal interference signal could bedisposed in each of the total of N_(LP) frequency bands. $\begin{matrix}{\quad{\begin{bmatrix}{{H\left( {{\overset{\_}{\omega}}_{normk} \cdot f_{0}} \right)}}^{2} & {{H\left( {{{- 2}\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right.}^{2}} & \cdots & {{H\left( {{{- \left( {N_{LP} - 1} \right)}\quad 2\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right)}}^{2} \\{{H\left( {{2\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right)}}^{2} & {{H\left( {{\overset{\_}{\omega}}_{normk} \cdot f_{0}} \right)}}^{2} & \cdots & {{H\left( {{{- \left( {N_{LP} - 2} \right)}\quad 2\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right)}}^{2} \\\vdots & \vdots & ⋰ & \vdots \\{{H\left( {{\left( {N_{LP} - 1} \right)\quad 2\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right)}}^{2} & {{H\left( {{\left( {N_{LP} - 2} \right)\quad 2\pi\quad f_{0}} + {{\overset{\_}{\omega}}_{normk} \cdot f_{0}}} \right)}}^{2} & \cdots & {{H\left( {{\overset{\_}{\omega}}_{normk} \cdot f_{0}} \right)}}^{2}\end{bmatrix} \cdot {\quad{\begin{bmatrix}{\hat{P}}_{n_{start}} \\{\hat{P}}_{n_{start} + 1} \\\vdots \\{\hat{P}}_{n_{start} + N_{LP} - 1}\end{bmatrix} = {\begin{bmatrix}P_{{MU},n_{start}} \\P_{{MU},{n_{start} + 1}} \\\vdots \\P_{{MU},{n_{start} + N_{LP} - 1}}\end{bmatrix} + \begin{bmatrix}e_{n_{start}} \\e_{n_{start} + 1} \\\vdots \\e_{n_{start} + N_{LP} - 1}\end{bmatrix}}}}}} & (30)\end{matrix}$

The equation system (30) can be presented in an equivalent manner viathe short form of equation (31).H·{circumflex over (P)}=P _(MU) +e   (31)

The error vector e=[e_(n) _(start) , e_(n) _(start) ₊₁, . . . ,e_(n)_(start) _(+N) _(LP) ⁻¹]^(T) represents the error of the MUSIC algorithmin the respective power-level curve. If the error vector e is set tozero, then an unambiguous solution is obtained for the estimated vector{circumflex over (P)} of the sought power level P_(k) according toequation (32):{circumflex over (P)}=H ⁻¹ ·P _(MU)  (32)

Through the error vector e, error power-level values {circumflex over(P)}_(k) can occur, some of which can also be negative. In a case ofthis kind, by resolving the optimisation problem according to equation(33), which guarantees a non-negative estimated vector {circumflex over(P)} for the sought power-level values P_(k), a better solution can beachieved than by resolving the equation system (32). $\begin{matrix}{{\min\limits_{\hat{p}}{{{{H \cdot \underset{\_}{\hat{P}}} - {\hat{\underset{\_}{P}}}_{MU}}}^{2}{under}\quad{subsidiary}\quad{condition}\quad\hat{p}}} > 0} & (33)\end{matrix}$

The frequency f_(k) of the spectral line found, for example, by means ofthe MUSIC algorithm, is determined according to equation (34) from thenumber ν_(Bin) of the associated frequency band (counted from 1 toN_(FFT)) with the addition of the weighted, scaled angular frequency{overscore (ω)}_(normk). $\begin{matrix}{f_{k} = {\left( {\left( {v_{Bin} - 1} \right) + \frac{{\overset{\_}{\omega}}_{normk}}{2\pi}} \right) \cdot f_{0}}} & (34)\end{matrix}$

If only one sideband—in the case of a real measured signal x(t)—isevaluated, then the respectively-determined power-level value{circumflex over (P)}_(k) must still be multiplied by the factor 2.

After all of the main lines and subsidiary lines, associated with all ofthe sinusoidal interference signals present in the noise signal havebeen identified, either a frequency and power level comparison of theidentified sinusoidal interference signals can be carried out withreference interference signals in the framework of an evaluation, or, asdescribed below, a removal according to the invention of the identifiedspectral lines associated with sinusoidal interference signals can beimplemented. Alternatively, spectral lines, of which the frequencies areknown, for example, in the case of the disturbance of the noise signalby system hum, can also be removed without the identification accordingto the invention described above using the method according to theinvention for the removal of sinusoidal interference signals from anoise signal.

The starting point is a list with frequencies f_(k), at which sinusoidalinterference signals occur. The removal of the spectral lines associatedwith the sinusoidal interference signals takes place, once again, byanalogy with the detection of interference-signal spectral lines, inseveral frequency bands generated by an FFT filter bank 1. The list cancontain positive frequencies f_(k) associated with real sinusoidalinterference signals or positive and negative frequencies f_(k)associated with complex interference-signal rotating phasors. Forreasons of simplicity, the removal of real interference signals will beconsidered below.

After determining the number L_(H) of frequency bands or respectivelyFFT bins, which are covered by a window function |H(f)|² (corresponding,for example, to the frequency range, in which the window function|H(f)|² exceeds a given threshold value), the variables p(ν=1, . . . ,N_(FFT)/2), which count the number of interference-signal spectral linesper frequency band ν, are initialised—when observing one sideband—withthe value zero for every frequency band from 1 up to N_(FFT)/2.

For every frequency f_(k), at which a spectral line can occur, a test ofthe Nyquist criterion is carried out as a selection criterion. If thefrequency f_(k) does not satisfy the Nyquist criterion (condition inequation (35)), then the frequency f_(k) is discarded and no longerpursued.f _(k) >f _(s)/2=1/(Δt·2)  (35)

Following this, the number ν_(center) of the FFT bin or the respectivefrequency band according to equation (36), in which the main line of thesinusoidal interference signal with the frequency f_(k), is disposed, isdetermined: $\begin{matrix}{v_{center} = {{{round}\quad\left( {\frac{f_{k}}{f_{s}} \cdot N_{FFT}} \right)} + 1}} & (36)\end{matrix}$

Starting from the determined number ν_(center) of the FFT bins or therespective frequency band, in which the main line of the sinusoidalinterference signal with the frequency f_(k) falls, the countingvariables p(ν) of those frequency bands according to equation (38),which are disposed within the window function in the frequency range,are incremented. $\begin{matrix}{{{p(v)} = {{{p(v)} + {1\quad{for}\quad{all}\quad v}} = {v_{center} - \frac{L_{H} - 1}{2}}}},\ldots\quad,{v_{center} + \frac{L_{H} - 1}{2}}} & (37)\end{matrix}$

If the list with frequencies f_(k), at which sinusoidal interferencesignals occur, contains an estimate for the difference of thepower-level value {circumflex over (P)}_(k) of the spectral line of thenoise level associated with the sinusoidal interference signal inaddition to the frequencies f_(k), the number L_(H) of the frequencybands or respectively FFT bins for those spectral lines, of which thepower levels {circumflex over (P)}_(k) are disposed only slightly abovethe noise level, could be reduced in order to cut down the calculationtime.

When considering both sidebands or one respective sideband, the estimate{circumflex over (R)}_(ν) of the autocorrelation matrix associated withthe frequency band ν is determined for each of the total of N_(FFT) orrespectively N_(FFT)/2 frequency bands, starting from the time-discreteoutput signal x_(ν)(μ·ov) of the FFT filter bank 1 associated with therespective frequency band ν.

According to equation (38), for the removal according to the inventionof sinusoidal interference signals in a noise signal, the dimension M(ν)of the estimate {circumflex over (R)}_(ν) of the autocorrelation matrixassociated with the frequency band ν must correspond at least to thenumber p(ν), identified above, of spectral lines per frequency band νwith the addition of the value 1:M(ν)≧p(ν)+1  (38)

In this manner, the dimension M(ν) of the estimate {circumflex over(R)}_(ν) of the autocorrelation matrix associated with a frequency bandν for the method according to the invention for the removal ofsinusoidal interference signals from a noise signal can generally beinterpreted as smaller than the dimension M of the estimate {circumflexover (R)}_(ν) for the method according to the invention for thedetection of sinusoidal interference signals in a noise signal accordingto equation (11).

Moreover, the averaging lengths N_(avg), in the case of thedetermination of the estimate {circumflex over (R)}_(ν) of theautocorrelation matrix associated with a frequency band ν, can bedesigned to be smaller for the removal of spectral lines by comparisonwith the detection of spectral lines.

The determination of the estimate {circumflex over (R)}_(ν) of theautocorrelation matrix associated with the frequency band ν for themethod according to the invention for the removal of spectral linescorresponds to the procedure in the method according to the inventionfor the identification of spectral lines according to equation (10).

Starting from the estimate {circumflex over (R)}_(ν) of theautocorrelation matrix associated with the respective frequency band ν,the noise power σ_(w,ν) associated with the frequency band ν is thendetermined.

If no spectral line is present within the frequency band ν−p(ν)=0—, thenthe noise-power spectrum Ŝ(ν) is derived from the averaging of the spurelements of the estimate {circumflex over (R)}_(ν) of theautocorrelation matrix associated with a frequency band ν according tothe equation (39): $\begin{matrix}{{\hat{S}(v)} = {\sigma_{w,v}^{2} = {\frac{1}{M}{\sum\limits_{k = 1}^{M}{{\hat{R}}_{v}\left( {k,k} \right)}}}}} & (39)\end{matrix}$

This corresponds to the Welch method for spectral estimation known fromthe prior art.

With at least one spectral line per frequency band ν−p(ν)>0—the total ofM eigenvalues λ₁, . . . ,λ_(M) of the M-dimensional estimate {circumflexover (R)}_(ν) of the autocorrelation matrix associated with a frequencyband ν is sorted via an analysis of eigenvalues of {circumflex over(R)}_(ν) and in ascending order of their value—λ₁≦λ₂≦ . . . ≦λ_(M).According to equation (40), the noise-power spectrum Ŝ(ν) is once againderived with reference to equation (14) from the averaging of the M−p(ν)lowest eigenvalues λ₁≦λ₂≦ . . . ≦λ_(M−p(ν)) of the estimate {circumflexover (R)}_(ν) of the autocorrelation matrix associated with thefrequency band ν, which correspond to the noise eigenvalues of thematrix {circumflex over (R)}_(ν): $\begin{matrix}{{\hat{S}(v)} = {\sigma_{w,v}^{2} = {\frac{1}{M - {p(v)}}{\sum\limits_{k = 1}^{M - {p{(v)}}}\lambda_{k}}}}} & (40)\end{matrix}$

On the basis of the mathematical background described above, thesubroutines for measurement of a noise-power spectrum, identification ofsinusoidal interference signals in a noise signal and removal ofsinusoidal interference signals from a noise signal and the methodaccording to the invention for noise measurement with the combinablesubroutines of measurement, identification and removal of sinusoidalinterference signals in a noise signal are described below.

The subroutine for the measurement of a noise-power spectrum accordingto FIG. 9 begins in procedural stage S10 with the splitting of acontinuous or time-discrete measured signal x(t) or x(μ·Δt), whichrepresents a noise signal w(t) or w(μ·Δt) with superimposed sinusoidalinterference signals A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾ or A_(k)·e^(j(μ·ω)^(k) ^(Δt+φ) ^(k) ⁾), by means of an FFT filter bank 1 according toequation (1) into a total of N_(FFT) measured signals, of which thefrequency spectrum is bandpass-filtered respectively to a frequency bandν. Each of these measured signals at the output of the FFT filter bank 1is generated by one FFT of the FFT filter bank 1, wherein the respectivebandpass-filtering of the measured signal is implemented via a windowing2 ₁,2 ₂, . . . ,2 _(NFFT) and subsequent down mixing 3 ₁,3 ₂, . . . ,3_(NFFT) via the FFT. The individual FFTs of the FFT filter bank 1 readin the respective measured signal at the input of the FFT filter bank 1with a time offset and therefore operate with a time overlap withreference to the measured signal at the input. The oversampling of theindividual FFT results caused because of this time overlap of theindividual FFTs according to equation (7) is compensated by acomplementary undersampling in subsequent undersamplings 4 ₁,4 ₂, . . .,4 _(NFFT). In order to generate a total of N_(FFT) measured signals ofa given frequency band ν respectively bandpass-filtered with regard totheir frequency spectrum at the output of the FFT filter bank 1, thebandwidth of the output signal of the individual FFTs is limitedrespectively to one FFT bin length f₀.

In the next procedural stage S20, starting from a total of N_(FFT)measured signals, respectively bandpass-filtered with reference to theirfrequency spectrum to a given frequency band ν, an estimate {circumflexover (R)}_(ν) of the autocorrelation matrix associated with eachfrequency band ν is determined according to equation (10). As a resultof the stochastic characteristic of the noise signal, in this context,the unbiased quality of the estimate {circumflex over (R)}_(ν) of therespective autocorrelation matrix is increased by multiple averaging.According to equation (10), the maximum dimension M of theautocorrelation matrices for the calculation of the noise-power spectrumŜ(ν·f₀) with superimposed spectral lines of sinusoidal interferencesignals associated with the respective frequency bands ν is reducedaccording to the Welch method to the value 1.

The subroutine according to the invention for the identification of theinterference-signal spectral lines in a noise-power spectrum accordingto FIG. 10 provides in its two first procedural stages S110 and S120identical procedural stages to the two procedural stages S10 and S20 ofthe subroutine for the measurement of a noise-power spectrum accordingto FIG. 9. The only difference relative to the subroutine for themeasurement of a noise-power spectrum is that the matrix dimension ofthe respective autocorrelation matrices {circumflex over (R)}_(ν) forthe subroutine for the identification of interference-signal spectrallines is specified as at least the maximum expected number P_(max) ofinterference-signal spectral lines per frequency band ν with theaddition of the value 2.

The next procedural stage S130 of the subroutine according to theinvention for the identification of interference-signal spectral linescomprises the determination of the eigenvalues λ₁, . . . ,λ_(M) and theassociated eigenvectors v ₁, . . . ,v _(M) of each of the total ofN_(FFT) estimates {circumflex over (R)}_(ν) of the autocorrelationmatrix associated with each frequency band ν, which is implementedaccording to known mathematical methods for eigenvalue analysis ofmatrices and the determination of corresponding eigenvectors based uponthese methods.

In the next procedural stage S140, for every frequency band ν andaccordingly for the estimate {circumflex over (R)}_(ν) of theautocorrelation matrix associated with each frequency band ν, asubdivision of all respectively determined eigenvalues λ₁, . . . ,λ_(M)into a first group of eigenvalues λ₁, . . . ,λ_(M−p(ν)) associated withthe noise components and into a second group of eigenvaluesλ_(M−p(ν)+1), . . . ,λ_(M) associated with the sinusoidal interferencesignal components is implemented. With the number of noise eigenvaluesλ₁, . . . ,λ_(M−p(ν)) and the number of interference-signal eigenvaluesλ_(M−p(ν)+1), . . . ,λ_(M), the number p(ν) of sinusoidal interferencesignals per frequency band ν is determined from procedural stage S140.

In the next procedural stage S150, the noise power σ_(w,ν) of eachfrequency band ν is calculated according to equation (14) with thedetermined noise eigenvalues.

In the next procedural stage S160, the scaled angular frequenciesω_(normk) of the sinusoidal interference signals disposed in therespective frequency band ν are determined by determining an estimationfunction P_(ν) associated with the respective frequency band ν, which isbased upon the eigenvalues and eigenvectors of the estimate {circumflexover (R)}_(ν) of the autocorrelation matrix associated with therespective frequency band ν, and by evaluating this estimation functionP_(ν). The MUSIC (Multiple Signal Classification) estimation functionP_(MU,ν) presented in equation (17) can be used, by way of example, asthe estimation function P_(ν). Alternatively, other estimation methodsbased upon the eigenvalue analysis of autocorrelation matrices can alsobe used. According to equation (17), the MUSIC estimation functionP_(MU,ν) provides maxima at those angular frequencies ω_(normk), atwhich respectively an eigenvector v _(i) associated with a noisecomponent is orthogonal to an arbitrary column vector e _(i) of thesignal autocorrelation matrix R_(s) and at which, accordingly, thescalar product in the denominator expression comprising respectively aneigenvector v _(i) associated with a noise component and an arbitrarycolumn vector e _(i) of the signal autocorrelation matrix R_(s) is zero.The scaled angular frequencies ω_(normk) of the sought sinusoidalinterference signals are derived from the scaled angular frequenciesω_(normk) of the largest maxima of the estimation function P_(MU,ν)corresponding to the number of interference-signal eigenvaluesλ_(M−p(ν)+1), . . . ,λ_(M).

In procedural stage S160, the determination of the power-level valuesP_(MU,ν,k) associated with the individual sinusoidal interferencesignals for each frequency band ν is additionally implemented byresolving the linear equation systems (20). For this purpose, therespective noise power σ_(w,ν), all of the interference-signaleigenvalues and the Z-transform V_(i)(e^(jω) ^(normk) ) of theeigenvectors λ_(M−p(ν)+1), . . . ,λ_(M) obtained from the individualinterference-signal eigenvalues v _(M−p(ν)+1), . . . ,v _(M) arerequired for each frequency band ν, wherein the individual Z-transformsV_(i)(e^(jω) ^(normk) ) calculated at the individual scaled angularfrequencies e^(jω) ^(normk) of the sinusoidal interference signals areobtained through the estimation function P_(MU,ν,k).

In the next procedural stage S170, all of the spectral lines with theirrespective power-level values P_(MU,ν,k), which were identified in theprevious procedural stage S160 in the individual frequency bands ν at anapproximately identical, scaled angular frequency ω_(normk), arecombined to form a combined power-level curve according to FIG. 7. Thedifference Δω_(normk) between the two scaled angular frequencies, which,according to condition (28), must be smaller than a specified maximumangular-frequency difference Δω_(normkMAx), is used as the criterion forapproximately-equal angular frequencies Δω_(normk) of spectral lines.For every scaled angular frequency Δω_(normk), the respectivepower-level curve is derived from the power-level values P_(MU,ν,k)localised at the individual FFT bin frequencies f₀, respectivelyassigned to the individual frequency bands ν.

The spectral lines respectively identified at a given scaled angularfrequency Δω_(normk) in the individual frequency bands ν in proceduralstage S160 can be derived respectively from a superimposition of severalspectral lines. This superimposition can result from main lines andsubsidiary lines of one or more sinusoidal interference signals, ofwhich the frequency is disposed in a frequency band ν, and at least onesubsidiary line of at least one further sinusoidal interference signal,of which the frequencies are disposed in frequency bands ν±i adjacent tothe frequency band ν and which come to be disposed in the frequency bandν as a result of the leakage effect.

In procedural stage S180, the power-level values {circumflex over(P)}_(k) of the individual spectral lines, which result from sinusoidalinterference signals with frequencies in different frequency bands ν±i,are determined by resolving the linear equation systems (30) and (33).The power-level values P_(MU,ν,k) disposed in the preceding proceduralstage S170 at the individual FFT bin frequencies ν·f₀ of the spectrallines of the respective sinusoidal interference signals, determined inprocedural stage S160, for example, using the MUSIC method, and themodulus-squared window transmission functions |H(f)|²frequency-displaced by the individual FFT bin frequencies ν·f₀ of theFFT filter bank 1 are entered in the linear equation systems (30) and(33), applicable for a respectively scaled angular frequency ω_(normk).Ignoring an error vector integrated in the equation systems (30) and(33), which models the process error achieved with the MUSIC method, theindividual power-level values {circumflex over (P)}_(k) of theindividual spectral lines are calculated by inversion of the matrix Hformed with the individual window-transmission functions |H(f)|² andsubsequent multiplication by the vector P_(MU,ν) from the power-levelvalues P_(MU,ν,k) determined by means of the MUSIC method in theindividual frequency bands ν according to equation (32). Alternatively,the individual power-level values {circumflex over (P)}_(k) of theindividual spectral lines can also be determined by minimising the errorvector e within the context of a minimisation method according toequation (33).

In the next procedural stage S190, the respective frequency f_(k) isdetermined for every spectral line according to equation (34).

In the final procedural stage S195, a list is prepared with all of thespectral lines at the frequencies f_(k), which are to be removed.

In the following section, the subroutine according to the invention forthe removal of sinusoidal interference signals from a noise signalaccording to FIG. 11 is described:

In the first procedural stage S210, the number p(ν) ofinterference-signal spectral lines to be removed is determined for everyfrequency band ν, starting from the list of all sinusoidal interferencesignals identified in the frequency range as prepared in proceduralstage S195 of the subroutine according to the invention for theidentification of sinusoidal interference signals. In this context, allspectral lines, of which the frequencies according to equation (35) donot satisfy the Nyquist condition and which are not useful for furtherprocessing are discarded. The frequency band ν_(Center), in which themain line of the sinusoidal interference signal comes to be disposed, iscalculated according to equation (36), in order to determine the numberp(ν) of interference-signal spectral lines per frequency band ν to beremoved, starting from the frequency f_(k) of the identified sinusoidalinterference signal. All frequency bands according to equation (37), inwhich respectively a main line or one of the subsidiary lines of thesinusoidal interference signal are disposed, are incremented on thisbasis.

In the next procedural stage S220, by analogy with procedural stage S10in the subroutine according to the invention for the measurement ofsinusoidal interference signals in a noise signal or respectivelyprocedural stage S110 of the subroutine for the identification ofsinusoidal interference signals in a noise signal, a total of N_(FFT)measured signals, of which the frequency spectra are bandpass-filteredrespectively with reference to one of the frequency bands ν, isdetermined. The total of N_(FFT) measured signals, of which thefrequency spectra are bandpass-filtered respectively with regard to oneof the frequency bands ν, is determined according to equation (1) via anFFT filter bank 1.

In the next procedural stage S230 of the subroutine according to theinvention for the removal of sinusoidal interference signals in a noisesignal, by analogy with procedural stage S20 of the subroutine for themeasurement of the noise-power spectrum or respectively procedural stageS120 of the subroutine according to the invention for the identificationof sinusoidal interference signals in a noise signal, a respectiveestimate {circumflex over (R)}_(ν) of an autocorrelation matrixassociated with each frequency band ν is calculated starting from thetotal of N_(FFT) measured signals, of which the frequency spectra arebandpass-filtered respectively with regard to one of the frequency bandsν according to equation (10). In this context, the matrix dimension M(ν)of the autocorrelation matrix {circumflex over (R)}_(ν) associated withthe frequency band ν is, however, adjusted at least to the number p(ν)of interference-signal spectral lines with the addition of the value 1.As a result of the stochastic characteristic of the noise signal, theunbiased quality of the estimate {circumflex over (R)}_(ν) of therespective autocorrelation matrix associated with each frequency band νis increased by multiple averaging.

The next procedural stage S240 contains the eigenvalue analysis, using aknown mathematical method, of each of the autocorrelation matrices,which are assigned to each frequency band ν.

With reference to the number p(ν) of noise eigenvalues determined inprocedural stage S210 for each frequency band ν, in procedural stageS250, the respective noise power σ_(w,ν) is calculated for eachfrequency band ν from the sum of the M(ν)−p(ν) smallest eigenvalues,wherein M(ν) is the dimension of the estimate {circumflex over (R)}_(ν)of the autocorrelation matrix associated with the frequency band ν.

In procedural stage S260, the total discrete noise-power spectrumŜ(ν·f₀) is determined from the noise powers σ_(w,ν) associated with eachof the individual frequency bands ν, which were determined in theprevious procedural stage S250.

In the following section, the first embodiment of the method accordingto the invention for noise measurement with the combinable subroutinesfor the measurement, identification and removal of sinusoidalinterference signals in a noise signal is described with reference toFIG. 12:

In the first embodiment of the method according to the invention fornoise measurement with the combinable subroutines for the measurement,identification and removal of sinusoidal interference signals in a noisesignal according to FIG. 12, the two procedural stages of analysis ofthe measured signal x(t) or x(ν·Δt) into a total of N_(FFT) measuredsignals, bandpass-filtered respectively to one of the N_(FFT) frequencybands and calculation of the autocorrelation matrix {circumflex over(R)}_(ν) respectively associated with a frequency band ν are calculatedonly once in procedural stages S10 and S20 for all three subroutines forthe measurement, identification and removal of interference-signalspectral lines in a noise-power spectrum. In this context, the matrixdimension M of the respective autocorrelation matrix {circumflex over(R)}_(ν) is specified according to equation (11) for all threesubroutines together at the maximum required value of the maximumexpected number P_(max) of interference-signal spectral lines perfrequency band ν with the addition of the value 2. For the calculationof the noise-power spectrum Ŝ(ν·f₀) in procedural stage S20 of thesubroutine for the measurement of the noise-power spectrum withsinusoidal interferers, only the mean value of the elements {circumflexover (R)}_(ν) (k,k) (k=1, . . . , M) of the respective autocorrelationmatrix {circumflex over (R)}_(ν) is taken.

After the calculation of the noise-power spectrum with sinusoidalinterferers in procedural stages S10 and S20, the identification ofsinusoidal interferers in the noise-power spectrum is implemented in theremaining required procedural stages S130 to S195 of the firstembodiment. (Procedural stages S110 and S120 of the subroutine accordingto the invention for the identification of sinusoidal interferers in anoise-power spectrum have already been processed in procedural stagesS10 and S20 of the subroutine for the measurement of the noise-powerspectrum with sinusoidal interferers).

In the case of a high noise variance of the measured signal x(t) orx(ν·Δt) and accordingly a problematic identification of the sinusoidalinterferers in the noise-power spectrum, in procedural stage S200, thenumber N_(avg) of the averagings of the estimate {circumflex over(R)}_(ν) of the repsective autocorrelation matrix to be implemented isincreased within the framework of an initialisation of the respectiveautocorrelation matrices and accordingly, the stochastic noise componentin the measured signal x(t) or x(ν·Δt) is minimised by comparison withthe deterministic interference-signal spectral lines, and the procedurefor measuring the noise-power spectrum is re-started in procedural stageS10.

Finally, after the identification of the sinusoidal interferers in thenoise-power spectrum according to procedural stages S130 to S195, theremoval of the identified sinusoidal interferers from the noise-powerspectrum is implemented in the remaining required stages S210, S240 toS260 of the first embodiment. The procedural stages S220 and S230 of thesubroutine according to the invention for the removal of sinusoidalinterferers from a noise-power spectrum have already been processed inprocedural stages S10 and S20 of the subroutine for the measurement ofthe noise-power spectrum with sinusoidal interferers.

After the removal of the sinusoidal interferers from the noise-powerspectrum, the measurement of the noise-power spectrum is re-startedcyclically in procedural stage S10.

The following section describes the second embodiment of the methodaccording to the invention for noise measurement with the combinablesubroutines for the measurement, identification and removal ofsinusoidal interference signals in a noise signal with reference to FIG.13.

In the second embodiment of the method according to the invention fornoise measurement with the combined subroutines for the measurement,identification and removal of sinusoidal interference signals in a noisesignal with reference to FIG. 13, subroutine-specific matrix dimensionsM for the autocorrelation matrices {circumflex over (R)}_(ν) associatedwith the respective frequency bands ν are used for each of the threesubroutines of measurement, identification and removal.

In the first procedural stages S10 and S20, which are identical to thoseof the subroutine for the measurement of the noise-power density in FIG.9, the autocorrelation matrices {circumflex over (R)}_(ν) each have thedimension M=1 and are therefore calculated in an efficient manner withregard to calculation time. In this manner, it is possible to calculatethe noise-power density in realtime quasi-continuously; this isindicated by the feedback branch to the start of the flow diagramadjoining procedural stage S20.

If the user of the noise measurement system intends to identify theoccuring interference-signal spectral lines, the autocorrelationmatrices {circumflex over (R)}_(ν) associated with the respectivefrequency bands ν are re-initialised with a dimension M, whichcorresponds to the maximum expected number P_(max) ofinterference-signal spectral lines per frequency band ν with theaddition of the value 2. The subroutine according to the invention forthe identification of sinusoidal interferers in a noise-power spectrumis implemented with the autocorrelation matrices {circumflex over(R)}_(ν) initialised with the new matrix dimension M in proceduralstages S110 to S195.

If the identification of the sinusoidal interferers in the noise-powerspectrum with the maximum matrix dimension M=P_(max)+2 for theindividual autocorrelation matrices proves difficult because of the highnoise variance, in procedural stage S200, by analogy with the firstembodiment in FIG. 12, the user can achieve a stronger averaging out ofthe stochastic noise-signal components in the measured signal andtherefore a more accurate identification of the interference-signalspectral lines by increasing the averaging number N_(avg) within theframework of an initialisation of the individual autocorrelationmactrices {circumflex over (R)}_(ν). The subroutine for theidentification of sinusoidal interferers in the noise-power spectrum isrepeated in procedural stage S110 with the new averaging number N_(avg).

If the subroutine for the identification of sinusoidal interferers in anoise-power spectrum is not repeated—which is the normal case—theindividual autocorrelation matrices {circumflex over (R)}_(ν) areinitialised in procedural stage S30 with the matrix dimension M=1 forthe measurement of the noise-power spectrum in procedural stages S10 andS20.

Following the implementation of the subroutine for the identification ofsinusoidal interferers in a noise-power spectrum, if the user intends toremove the identified sinusoidal interferers from the noise-powerspectrum, the autocorrelation matrix {circumflex over (R)}_(ν)associated with the respective frequency band ν is initialised with therespective matrix dimension M(ν), which corresponds to the number p(ν)of sinusoidal interferers identified in the frequency band ν with theaddition of the value 1.

The subroutine for the removal of the sinusoidal interferers in anoise-power spectrum is then implemented with the re-initialisedautocorrelation matrices {circumflex over (R)}_(ν) in procedural stagesS210 to S260. This subroutine can be implemented by the user severaltimes—which is indicated by the feedback branch from the end ofprocedural stage S260 to the beginning of procedural stage S210—becausethe calculations are less calculation-intensive by comparison with thesubroutine for the identification of sinusoidal interferers as a resultof the matrix dimension.

If the user does not wish to repeat the removal of identified sinusoidalinterferers, the individual autocorrelation matrices {circumflex over(R)}_(ν) are initialised in procedural stage S30 with the matrixdimension M=1 for the measurement of the noise-power spectrum inprocedural stages S10 and S20.

The following section describes the system according to the inventionfor noise measurement with the combinable subroutines for themeasurement, identification and removal of sinusoidal interferencesignals in a noise signal with reference to FIG. 14.

The continuous or time-discrete measured signal x(t) or x(ν·Δt) consistsof a continuous or time-discrete noise signal w(t) or w(ν·Δt) andseveral continuous or time-discrete sinusoidal interference signalsA_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾ or A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k)⁾). This continuous or time-discrete measured signal x(t) or x(ν·Δt) isprocessed in an FFT filter bank 1, of which the structure is presentedin detail in FIG. 3, in a total of N_(FFT) overlapping FFTs. Therespective frequency spectra of the continuous and time-discretemeasured signal x(t) or x(ν·Δt) are bandpass-filtered with regard to agiven frequency band ν in the windowings 2 ₁,2 ₂, . . . ,2 _(NFFT)assigned to the individual FFTs and in the subsequent down mixings 3 ₁,3₂, . . . ,3 _(NFFT). The individual measured signals, of which thefrequency spectrum is bandpass-filtered with regard to a frequency bandν, are then restored in subsequent undersamplings 4 ₁,4 ₂, . . . ,4_(NFFT), with regard to the oversampling caused by the overlapping, tothe original sampling rate according to FIG. 3.

The measured signals x(ν·ov·Δt), of which the respective frequencyspectrum is bandpass-filtered with regard to a frequency band ν, aresupplied at the total of N_(FFT) outputs of the FFT filter bank 1, tothe units 5 ₁,5 ₂, . . . ,5 _(NFFT) for the estimation of theautocorrelation matrix {circumflex over (R)}_(ν). The individualeigenvalues and eigenvectors of the estimate {circumflex over (R)}_(ν)of the autocorrelation matrix associated with each frequency band ν areadditionally determined in the respective units 5 ₁,5 ₂, . . . ,5_(NFFT) for the estimation of the autocorrelation matrix {circumflexover (R)}_(ν), from the estimate {circumflex over (R)}_(ν) associatedwith each frequency band ν of the autocorrelation matrix.

In a subsequent unit 6 ₁,6 ₂, . . . ,6 _(NFFT) for determining thenumber p(ν) of interference signals per frequency band ν, a separationof all eigenvalues λ₁, . . . ,λ_(M) determined at the estimated value{circumflex over (R)}_(ν) of the autocorrelation matrix associated withthe frequency band ν into noise eigenvalues λ₁, . . . . ,λ_(M) andinterference eigenvalues λ₁, . . . ,λ_(M−p(ν)) takes place. The numberp(ν) of interference signals per frequency band ν corresponds to thenumber of determined interference-signal eigenvalues λ_(M−p(ν)+1), . . .,λ_(M) of the estimate {circumflex over (R)}_(ν) associated with thefrequency band ν of the autocorrelation matrix.

The subsequent units 7 ₁,7 ₂, . . . ,λ_(NFFT) determine respectively thenoise power σ_(w,ν) of the frequency band ν from the noise eigenvaluesλ₁, . . . ,λ_(M−p(ν)), which corresponds respectively to a given numberof smallest eigenvalues determined in the units 5 ₁,5 ₂, . . . ,5_(NFFT) for the estimation of the autocorrelation matrix {circumflexover (R)}_(ν), wherein the given number represents the dimension Mreduced by the number p(ν) of interference signals per frequency band νof the estimate {circumflex over (R)}_(ν) associated with the frequencyband ν of the autocorrelation matrix.

The subsequent units 8 ₁,8 ₂, . . . ,8 _(NFFT) for frequency estimationdetermine, for example, via the MUSIC method using an estimationfunction by means of a maximum value observation, the scaled angularfrequencies ω_(normk) of the interference-signal spectral linesoccurring in the respective frequency band ν. In this context, themaxima occur in those cases, in which the eigenvectors v ₁, . . . ,v_(M−p(ν)) associated with the respective noise eigenvalues λ₁, . . . ,λ_(M−p(ν)) are orthogonal to an arbitrary column vector e _(i) of thesignal autocorrelation matrix R_(s) of the respective frequency band ν.

Finally, the power-level values P_(MU,ν,k) associated with theindividual interference-signal spectral lines in the individualfrequency bands ν are determined in subsequent units 9 ₁,9 ₂, . . . ,9_(NFFT) for power-level measurement, starting from the noise powersν_(w,ν), the interference-signal eigenvalues λ_(M−p(ν)+1), . . . , λ_(M)and the Z-transforms V_(u)(e^(jω) ^(normk) ) of the interference-signaleigenvectors v _(M−p(ν)+1), . . . ,v _(M) at the individual determined,scaled angular frequencies ω_(normk) of the individualinterference-signal spectral lines.

In a subsequent unit 10 for the measurement of power-level curves perdetermined, scaled angular frequency ω_(normk), all of the power-levelvalues P_(MU,ν,k) of spectral lines, which provide approximatelyidentical scaled angular frequencies ω_(normk) in different frequencybands ν, are combined at the individual FFT bin frequencies ν·f₀ of theFFT filter bank 1 to form a power-level curve.

The power-level values {circumflex over (P)}_(k) of individual spectrallines, which result from different sinusoidal interference signals withfrequencies f_(k) in different frequency bands ν and which aresuperimposed at an identical scaled angular frequency ω_(normk) to forma single spectral line, are determined in a unit 11 for determiningindividual spectral lines from a superimposed spectral line. Thissuperimposition of individual spectral lines, which is based upon theoccurrence of subsidiary lines of sinusoidal interference signalsbecause of leakage effects caused by windowing in respectively adjacentfrequency bands ν and the superposition, accurate with regard tofrequency, of main lines of sinusoidal interference signals, is reversedby a deconvolution. For this purpose, a linear equation system with amatrix H comprising the modulus-squared window transmission functions|H(f−ν·f₀)|² frequency-displaced by the respective FFT bin frequencyν·f₀, the vector P _(MU,ν) with the power-level values P_(MU,ν,k)determined at the respective scaled angular frequency ω_(normk) in theindividual adjacent frequency bands in the units 9 ₁,9 ₂, . . . ,9_(NFFT) for power-level measurement and the vector {circumflex over (P)}of the sought power-level values {circumflex over (P)}_(k) of theassociated individual spectral lines, is resolved.

After a determination of the frequencies f_(k) associated with thepower-level values {circumflex over (P)}_(k) of the determined spectrallines in the unit 11 for determining individual spectral lines from asuperimposed spectral line, the individual, identified spectral lineswith their respective power-level values {circumflex over (P)}_(k) andfrequencies f_(k) are entered in respective lists in a unit 14.Alternatively, in unit 14, the power-level values P_(k) and frequenciesf_(k) of spectral lines in a noise spectrum to be removed can be enteredin the respective lists from an external source.

In order to remove the identified or specified spectral lines ofsinusoidal interference signals, the eigenvalues λ₁, . . . ,λ_(M)determined in the total of N_(FFT) estimation units 5 ₁, 5 ₂, . . . , 5_(NFFT) of the autocorrelation matrices {circumflex over (R)}_(ν)associated with the respective frequency bands ν are supplied to thetotal of N_(FFT) units 7 ₁′,7 ₂′, . . . ,7 _(NFFT)′ for thedetermination of the noise power σ_(w,ν) associated with the respectivefrequency band ν. These N_(FFT) units 7 ₁′,7 ₂′, . . . ,7 _(NFFT)′ forthe determination of the noise power σ_(w,ν) associated with therespective frequency band ν are supplied by a unit 12 for determiningthe number of sinusoidal interference signals per frequency band νrespectively with the number p(ν) of sinusoidal interference signals perfrequency band ν. The unit 12 for determining the number of sinusoidalinterference signals per frequency band ν evaluates the list prepared inunit 14 with all of the identified or specified spectral lines ofsinusoidal interference signals.

The individual units 7 ₁′, 7 ₂′, . . . , 7 _(NFFT)′ for determining thenoise powers ν_(w,ν) associated with the individual frequency bands νare connected to a unit 13 for generating the noise-power spectrum Ŝ(ν),in which the noise-power spectrum Ŝ(ν) of the entire frequency range tobe measured is determined without the undesired spectral lines of thesinusoidal interference signals.

Finally, the unit 15 generates the noise-power spectrum extending overthe entire frequency range including the spectral lines of sinusoidalinterference signals from noise spectra including the spectral lines ofsinusoidal interference signals determined in the units 5 ₁, 5 ₂, . . ., 5 _(NFFT) for the estimation of the autocorrelation matrix {circumflexover (R)}_(ν) for the respective frequency band ν. In order to determinethe noise-power spectrum in the units 5 ₁, 5 ₂, . . . , 5 _(NFFT) forthe estimation of the autocorrelation matrix {circumflex over (R)}_(ν),the dimension M of the respective autocorrelation matrix {circumflexover (R)}_(ν) is set to the value 1, so that the noise-power spectrumassociated with the respective frequency band ν including the spectrallines of sinusoidal interference signals is derived in the sense ofequation (12), from the calculation of an autocorrelation matrix{circumflex over (R)}_(ν) of this kind.

FIG. 15 presents a phase-noise spectrum with superimposed spectral linesof sinusoidal interference signals. The display also contains thefrequencies at which spectral lines of sinusoidal interference signalshave been identified by the subroutine according to the invention forthe identification of sinusoidal interference signals in a noise signal.

FIG. 16 shows the identical phase-noise spectrum to that shown in FIG.15, from which spectral lines associated with sinusoidal interferencesignals have been removed by the subroutine according to the inventionfor the removal of sinusoidal interference signals in a noise signal.

The determination of the estimated value {circumflex over (R)}_(ν) forthe autocorrelation matrix associated with the frequency band ν in thecase of the method according to the invention for identification isgenerally completed after a total of N_(avg) averagings, and isimplemented continuously in the case of the method according to theinvention for the removal of interference-signal spectral lines, whereinthe removal of the spectral lines can begin after the provision of asingle estimate {circumflex over (R)}_(ν) of the autocorrelation matrixassociated with the frequency band ν, the accuracy of which can beimproved with increasing averaging lengths N_(avg) of the estimate{circumflex over (R)}_(ν) of the autocorrelation matrix associated withthe frequency band ν.

The invention is not restricted to the embodiments presented. Inparticular, instead of the MUSIC method, other frequency estimationmethods, which are based upon an eigenvalue analysis of autocorrelationmatrices, such as the Pisareko method can also be used.

1. Method with subroutines for the measurement, identification andremoval of sinusoidal interference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ)^(k) ⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾) in a noise signal (w(t),w(μ·Δt)), which can be combined in a system for noise measurement,wherein the procedural stage (S10, S110, S220) of splitting of thefrequency range (ν) to be measured into several frequency bands (ν) viaan FFT filter bank (1) and the procedural stage (S20, S120, S230) ofdetermining autocorrelation matrices ({circumflex over (R)}_(ν))respectively associated with the frequency bands (ν) is implemented in acombined manner for the respectively selected subroutines ofmeasurement, identification and removal of sinusoidal interferencesignals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ)^(k) ⁾) in a noise signal (w(t), w(μ·Δt)), and wherein the parameters ofthe autocorrelation matrices ({circumflex over (R)}_(ν)) are adjusted ina variable manner dependent upon the respectively selected subroutine(s)and upon the required quality of the result.
 2. Method according toclaim 1, wherein the parameters for the autocorrelation matrices({circumflex over (R)}_(ν)) are the number of averagings (N_(avg)) andthe matrix dimension (M) of the autocorrelation matrices ({circumflexover (R)}_(ν)).
 3. Method according to claim 2, wherein the adjustmentof the variable parameters of the autocorrelation matrices ({circumflexover (R)}_(ν)) takes place online within the framework of are-initialisation of the autocorrelation matrices ({circumflex over(R)}_(ν)).
 4. Method according to claim 2, wherein the matrix dimension(M) of the autocorrelation matrices ({circumflex over (R)}_(ν)) is equalto the value 1, if only the subroutine for the measurement of sinusoidalinterference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾, A_(k)·e^(j(μ·ω)^(k) ^(Δt+φ) ^(k) ⁾) in a noise signal (w(t), w(μ·Δt)) is implemented.5. Method according to claim 2, wherein the matrix dimension (M) of therespective autocorrelation matrix ({circumflex over (R)}_(ν)) is atleast the maximum expected number (P_(max)) of sinusoidal interferencesignals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ)^(k) ⁾) in the respective frequency band (ν) with the addition of thevalue 2, if only the subroutines for the measurement and identificationof the sinusoidal interference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k)⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾) in a noise signal (w(t),w(μ·Δt)) are implemented.
 6. Method according to claim 2, wherein thematrix dimension (M) of the respective autocorrelation matrix({circumflex over (R)}_(ν)) is at least the identified number (P_(max))of sinusoidal interference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾,A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾) in the respective frequency band(ν) with the addition of the value 1, if only the subroutines for themeasurement and removal of the sinusoidal interference signals(A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾)in a noise signal (w(t), w(μ·Δt)) are implemented.
 7. Method accordingto claim 2, wherein the matrix dimension (M) of the respectiveautocorrelation matrix ({circumflex over (R)}_(ν)) is at least themaximum expected number (P_(max)) of sinusoidal interference signals(A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾, A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾)in the respective frequency band (ν) with the addition of the value 2,if the subroutines for the measurement, identification and removal ofthe sinusoidal interference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾,A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾) in a noise signal (w(t), w(μ·Δt))are implemented at the same time.
 8. Method according to claim 2,wherein the number of averagings (N_(avg)) is increased in order toimprove the result quality.
 9. System for noise measurement with thecombinable subroutines for the measurement, identification and removalof sinusoidal interference signals (A_(k)·e^(j(ω) ^(k) ^(t+φ) ^(k) ⁾,A_(k)·e^(j(μ·ω) ^(k) ^(Δt+φ) ^(k) ⁾) in a noise signal (w(t), w(μ·Δt))with an FFT filter bank (1) for splitting a frequency range to bemeasured into several frequency bands (ν), which is shared for eachsubroutine, and respectively a unit (5 ₁, 5 ₂, . . . , 5 _(NFFT)) forthe calculation of an autocorrelation matrix ({circumflex over (R)}_(ν))associated with the each frequency band (ν), which is shared for eachsubroutine and of which the parameters can be adjusted dependent uponthe required subroutine(s) and attainable quality of the result. 10.System for noise measurement according to claim 9, wherein theparameters of the respective autocorrelation matrix ({circumflex over(R)}_(ν)) are the number of averagings (N_(avg)) and the matrixdimension (M) of the respective autocorrelation matrix ({circumflex over(R)}_(ν)).